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  "body": "\n## Definition\n\nLet $T$ be a recursively axiomatized theory that contains enough arithmetic to speak about [[computable|KZSB5E471K]] relations on $\\N$.\nFor a computable binary relation $\\prec$ on $\\N$, let $\\mathrm{WF}(\\prec)$ be the statement in the language of $T$ that $\\prec$ is a [[well-order|0F1D6PK9RZ]].\nFor first-order theories, $\\mathrm{WF}(\\prec)$ is replaced by the scheme of [[transfinite induction|0ZHQMQ2C1N]] along $\\prec$ for all formulas.\nThe *proof-theoretic ordinal* of $T$ is written $\\abs{T}$.\nIt is the supremum of the order types of all computable well-orders $\\prec$ on $\\N$ such that $T$ proves $\\mathrm{WF}(\\prec)$.\n\n## Remarks\n\nExample: for [[Peano arithmetic|0QFT7Z76N5]], $\\abs{\\mathrm{PA}} = \\varepsilon_0$, the ordinal [[epsilon nought|KT50SMH3EC]].\nGentzen showed that PA proves transfinite induction along a standard notation system for each ordinal below $\\varepsilon_0$, and not along the one for $\\varepsilon_0$ itself.\nNon-example: the least uncountable [[ordinal|0RDJK7EQYY]] $\\omega_1$ is not the proof-theoretic ordinal of any such theory, because a computable well-order on $\\N$ has countable order type.\nSuppose that every computable order that $T$ proves well-founded is in fact well-founded.\nThen $\\abs{T}$ is below the Church–Kleene ordinal $\\omega_1^{\\mathrm{CK}}$, the least ordinal that is not the order type of a computable well-order.\nSeveral inequivalent definitions are in use, for example with primitive recursive in place of computable orders, or with $\\Pi^0_2$ consequences in place of well-foundedness.\nFor the usual theories they give the same ordinal.\n",
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